Windowing methods and systems for use in time-frequency analysis

ABSTRACT

The present embodiments include methods of time-frequency analyzing signals. Some embodiments provide methods of processing signals comprising: receiving a communication signal; adaptively partitioning the signal in a time domain into a plurality of windows of the signal; transforming each of the windows of the signal producing respective expansions in a frequency domain and obtaining respective samples of the windows of signal in the frequency domain; and mapping the samples in the frequency domain back into the time domain.

This application claims the benefit of U.S. Provisional Application No.61/482,424, filed May 4, 2011, for Stephen D. Casey, entitled WINDOWINGSYSTEMS FOR TIME-FREQUENCY ANALYSIS, and U.S. Provisional ApplicationNo. 61/482,435, filed May 4, 2011, for Stephen D. Casey et al., entitledADAPTIVE AND ULTRA-WIDEBAND SAMPLING VIA PROJECTION, both of which areincorporated in their entirety herein by reference.

The United States Government may have rights in this invention pursuantto Contract No. DAAD19-02-D-0001 between U.S. Army Research OfficeScientific Services program and American University.

BACKGROUND

1. Field of the Invention

The present invention relates generally to time-frequency analysis, andmore specifically to signal processing.

2. Discussion of the Related Art

The communication of information has always been important. The amountof information that is communicated continues to rapidly increase.Further, the importance of those communications continues to increase.

As people become more mobile and as technology continues to advance theamount of information communicate is expected to continue to increase.

SUMMARY OF THE INVENTION

Several embodiments of the invention advantageously address the needsabove as well as other needs through methods of providing time-frequencyanalysis, such as in the processing of signals. Some embodiments providemethods of processing signals comprising: receiving a communicationsignal; adaptively partitioning the signal in a time domain into aplurality of windows of the signal; transforming each of the windows ofthe signal producing respective expansions in a frequency domain andobtaining respective samples of the windows of signal in the frequencydomain while preserving orthogonality of basis elements in the windows,including regions of overlap; and mapping the samples in the frequencydomain back into the time domain.

Further embodiments provide methods of processing a signal, comprising:receiving a signal; partitioning the signal in a time domain into aplurality of windows of the signal; and transforming each of the windowsof the signal producing respective expansions in a frequency domain,where for each window of the signal of the respective expansions areobtained through parallel processing obtaining in parallel respectivesamples of the windows of signal in the frequency domain.

Additionally, some embodiments provide method of processing a signal,comprising: processing a signal; partitioning the signal in a timedomain into a plurality of windows of the signal; and transforming eachof the windows of the signal producing respective expansions in afrequency domain and obtaining respective samples of the windows ofsignal in the frequency domain while preserving orthogonality between atleast two of the plurality of windows.

Other embodiments provide methods of processing a signal, comprising:receiving a communication signal; adaptively partitioning the signal ina time domain into a plurality of windows of the signal, wherein theadaptively partitioning comprises applying B-splines in constructing thewindows of the signal; and transforming each of the windows of thesignal producing respective expansions in a frequency domain andanalyzing the transformed windows of the signal in the frequency domain.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other aspects, features and advantages of severalembodiments of the present invention will be more apparent from thefollowing more particular description thereof, presented in conjunctionwith the following drawings.

FIG. 1 depicts a simplified flow diagram of a process of signal samplingand/or analyzing in accordance with some embodiments.

FIG. 2 depicts a simplified flow diagram of a process of projectionmethods of sampling according to some embodiments.

FIG. 3A depicts a simplified graphical representation of a signal.

FIG. 3B shows an enlarged version of a portion of the signal of FIG. 3Awith windows defined over a portion of the signal in accordance withsome embodiments.

FIG. 4 shows a graphical representation of a series of windows, inaccordance with some embodiments.

FIG. 5 shows an example sampling technique in accordance with a standardShannon (W-K-S) sampling.

FIGS. 6A-6B show simplified graphical representations of a method ofprojection sampling in accordance with some embodiments.

FIG. 7 shows a simplified graphical representation of at least portionsof a series of windows, in accordance with some embodiments, provided inresponse to the windowing provided in the process of FIG. 2.

FIG. 8 shows a simplified flow diagram of a process of providing thetransmission and/or analysis of the windowed signal, in accordance withsome embodiments.

FIG. 9 depicts a simplified flow diagram of a process of providing areconstruction and/or synthesis of coefficients in accordance with someembodiments.

FIG. 10 illustrates a system that may be used in processing signals inaccordance with at least some embodiments.

FIG. 11 illustrates a system for use in implementing methods,techniques, devices, apparatuses, systems, modules, units and the likein providing user interactive virtual environments in accordance withsome embodiments.

Corresponding reference characters indicate corresponding componentsthroughout the several views of the drawings. Skilled artisans willappreciate that elements in the figures are illustrated for simplicityand clarity and have not necessarily been drawn to scale. For example,the dimensions of some of the elements in the figures may be exaggeratedrelative to other elements to help to improve understanding of variousembodiments of the present invention. Also, common but well-understoodelements that are useful or necessary in a commercially feasibleembodiment are often not depicted in order to facilitate a lessobstructed view of these various embodiments of the present invention.

DETAILED DESCRIPTION

The following description is not to be taken in a limiting sense, but ismade merely for the purpose of describing the general principles ofexemplary embodiments. The scope of the invention should be determinedwith reference to the claims.

Reference throughout this specification to “one embodiment,” “anembodiment,” “some embodiments,” “some implementations” or similarlanguage means that a particular feature, structure, or characteristicdescribed in connection with the embodiment is included in at least oneembodiment of the present invention. Thus, appearances of the phrases“in one embodiment,” “in an embodiment,” “in some embodiments,” andsimilar language throughout this specification may, but do notnecessarily, all refer to the same embodiment.

Furthermore, the described features, structures, or characteristics ofthe invention may be combined in any suitable manner in one or moreembodiments. In the following description, numerous specific details areprovided, such as examples of programming, software modules, userselections, network transactions, database queries, database structures,hardware modules, hardware circuits, hardware chips, etc., to provide athorough understanding of embodiments of the invention. One skilled inthe relevant art will recognize, however, that the invention can bepracticed without one or more of the specific details, or with othermethods, components, materials, and so forth. In other instances,well-known structures, materials, or operations are not shown ordescribed in detail to avoid obscuring aspects of the invention.

The present embodiments provide methods and systems for use providingtime-frequency analysis, such as in processing communication signals andother relevant signals. The signal processing can include, for example,improved sampling, filtering, encoding, analog-to-digital conversion,and other processing. Some embodiments employ a windowing technique thatprovides effective time and/or frequency analysis of functions. Stillfurther, some embodiments provide windowing that can have variablepartitioning length, variable roll-off and/or variable smoothness. Thisvariable windowing can be particularly effective, for example, withadaptive frequency band (AFB) and ultra-wide band (UWB) signals andsystems, and other relevant signals and systems.

FIG. 1 depicts a simplified flow diagram of a process 110 of signalsampling and/or analyzing in accordance with some embodiments. In step112 a signal is received. In step 114, the signal is partitioned intowindows or blocks (sometimes referred to as tiles) based on time.Further, the partitioning can provide for adaptive partitioning suchthat windows or blocks can be can vary, in at least some embodiments,based on variations in, for example, bandwidth of the portions of thesignal relative to corresponding windows of time. In step 116, thewindows are processed to provide determine respective periodicexpansions, for example, through Fourier series. In manyimplementations, the respective periodic expansions are approximationsbased on the series. In step 120, sampling is performed of the window ofthe signal in the frequency domain relative to basis coefficients. Instep 122, the sampling is mapped back to the time domain. The mapping,in some instances can include reconstructing the signal from thesampling of basis coefficients to recover samples in the time domain.

The use of the varying blocks allows the methods and systems toaccommodate varying bandwidths within a signal. Further, this variationprovides enhanced sampling of varying signals, including signals such asadaptive frequency band (AFB) and ultra-wide band (UWB) signals.Additionally, some embodiments provide a sampling of signals through abasis projection as further describe below.

FIG. 2 depicts a simplified flow diagram of a process 210 of projectionmethods of sampling according to some embodiments. In step 212, a signalis windowed or blocked based on time. Again, the windowing may providefor adaptive windowing such that the windows can vary over the signal.In step 214, a time and/or frequency analysis is performed of the signalbased on the windowing and the sampling obtained through the windowing.In step 216, a synthesis of the sampling is performed, which in part canprovide a reindexing for the analysis of step 214.

FIG. 3A depicts a simplified graphical representation of a signal 310.The signal has a varying bandwidth, which can include for example a highfrequency bursts 312 and/or portions of low bandwidth variation. FIG. 3Bshows an enlarged version of a portion of the signal 310 of FIG. 3A,including the high frequency burst 312, with windows defined over aportion of the signal in accordance with some embodiments. As describedabove, some embodiments provide for windowing over the signal, and thewindows can span various time durations of the signal.

Still referring to FIG. 3B, multiple windows W₁-W_(k) are defined overthe portion of the signal 310 depicted. The windows W₁-W_(k) vary overthe signal. For example, the windows W₂, W₃, etc. corresponding to thehigh frequency burst 312 have periods τ₂, τ₃, etc. that are shorter intime than, for example, the period τ₁ of the window W₁ preceding thehigh frequency burst 312 of the signal 310. Some embodiments may providefor discontinuity between windows. Other embodiments, as furtherdescribed below, provide windows that can be defined with overlappingboundaries and/or regions.

The process of windowing a signal in time and/or frequency can enhancethe time-frequency analysis of a signal. Low-pass, high-pass orband-pass filters can be developed by effective windowing in frequency,whereas windowing systems in time provide tools for local analysis of agiven function. The present embodiments provide windowing methods andsystems that provide effective blocking (or tiling), efficient roll-off,and smoothness-decay. Further, some embodiments provide variablepartitioning length of the blocks, variable roll-off and/or variablesmoothness.

Some embodiments construct smooth adaptive partitions of unity, orbounded adaptive partitions of unity systems, which can begeneralizations of bounded uniform partitions of unity in that theyallow for signal adaptive windowing. This can give a flexible adaptivepartition of unity of variable smoothness and are useful whenever apartition of unity is used, such as in compressed sensing.

Many embodiments utilize splines, such as B-splines, as the constructionelements. Some embodiments additionally preserve orthogonality oforthonormal (ON) system between adjacent blocks. The construction hereuses orthonormal basis for L²(R) and can be created, for example, bysolving a Hermite interpolation problem with constraints. Theseorthonormal preserving windowing provides systems and methods oftime-frequency analysis for a wide class of signals. Alternatively, someembodiments employ a method of almost orthogonality. The almostorthogonality can employ the B-spline techniques to create almostorthogonal windowing that can, in at least some instances, be moreeasily computable and/or more readily implemented through hardwareand/or software.

In accordance with some embodiments, windowing for partitions of unity {

_(k)(t)} are configured with the intent of satisfying:

${\sum\limits_{k}{_{k}(t)}} \equiv 1.$

A difference between the partition of unity systems and orthonormalmethods and systems is that the orthonormal methods preserveorthogonality. Preserving orthogonality typically provides that thewindowing systems {

_(k)(t)} satisfy

[

_(k)(t))]²+[

_(k+1)(t))]²=1 for all k.

The almost orthogonal systems provide that there exists a δ, 0≦δ≦½ suchthat

1−δ≦[

_(k)(t))]²+[

_(k+1)(t))]²≦1+δ.

Accordingly, the present embodiments comprise methods and systems thatprovide computable atomic decomposition of time-frequency space relativeto signals and/or communications. At least some of these embodiments areconfigured to analyze signals and allow for changing frequency bandsand/or ultra-wide frequency bands. This can be achieved throughnon-uniformly windowing or blocking time and/or frequency. When a signalhas, for example, a burst of high-frequency information, someembodiments window or block quickly (or provide fairly short durationblocks) and efficiently in time and broadly in frequency, whereas whenthe signal has a relatively low-frequency segment, the methods andsystems can block or window broadly in time and efficiently infrequency. The methods can be implemented through hardware and/orsoftware.

At least some of the embodiments employ Fourier Series or other forms oftransformation. Accordingly, let ƒ be a periodic, integrable function on

, with period 2Φ. i.e., ƒεL¹(

_(2Φ)). The Fourier coefficients of ƒ. {circumflex over (ƒ)}[n], aredefined by:

${\hat{f}\lbrack n\rbrack} = {\frac{1}{2\Phi}{\int_{- \Phi}^{\Phi}{{f(t)}{\exp \left( {{- }\ \pi \; n\; {t/\Phi}} \right)}{{t}.}}}}$

If {{circumflex over (ƒ)}[n]} is absolutely summable ({{circumflex over(ƒ)}[n]}εl¹), then the Fourier series of ƒ can be defined by:

${f(t)} = {\sum\limits_{n \in {\mathbb{Z}}}{{\hat{f}\lbrack n\rbrack}{{\exp \left( {\; \pi \; n\; {t/\Phi}} \right)}.}}}$

For ƒεL¹, the Fourier transform {circumflex over (ƒ)}(w) is given asfollows.

With the Fourier transform and inversion formulae, let ƒεL¹. The Fouriertransform of ƒ can be defined as:

{circumflex over (ƒ)}(ω)=

ƒ(t)e^(−2πitω) dt,

for tε

(time), ωε

(frequency). The inversion formula, for {circumflex over (ƒ)}εL¹(

), can be defined by:

ƒ(t)=({circumflex over (ƒ)})^(V)(t)=

{circumflex over (ƒ)}(ω))e^(2πiωt) dω

The choice to have 2π in the exponent can simplify certain expressionsin accordance with some embodiments, e.g., Parseval's equality—

∥ƒ∥_(L) ₂ ₍

_()=∥{circumflex over (ƒ)}∥) _(L) ₂ ₍

₎.

The transform and the coefficient integral can be considered asanalysis, and the inverse transform and series as synthesis.

The process of periodization can be used in many if not all of theembodiments. Considering a block of a signal on an interval [0, T], theperiodization of that signal block can be used to expand a Fourierseries of sines and cosines. (see for example, J. J. Benedetto, HarmonicAnalysis and Applications, CRC Press, Boca Raton, Fla., 1997. [1, p.254], which is incorporated herein by reference in its entirety).

Further, letting T>0 and letting g(t) be a function such that supp g

[0, T]. The T-periodization of g can be:

${\lbrack g\rbrack^{\circ}(t)} = {\sum\limits_{n = {- \infty}}^{\infty}{{g\left( {t - {n\; T}} \right)}.}}$

Classical sampling theory applies to band-limited square integrablefunctions. A function that is both band-limited and L² typically hasseveral smoothness and growth properties given in the Paley-WeinerTheorem. This class of functions can be referred to as

_(Ω). The Whittaker-Kotel'nikov-Shannon (W-K-S) Sampling Theorem appliesto functions in

Ω. The Paley-Wiener Space

_(Ω) can be defined as:

_(Ω) ={ƒ: ƒ:{circumflex over (ƒ)}εL ²; supp({circumflex over (ƒ)}⊂[−Ω,Ω]}

Based on the W-K-S Sampling Theorem, and let

${f \in {\mathbb{P}}_{\Omega}},{{\sin \; {c_{T}(t)}} = {{\frac{\sin \left( {\frac{\pi}{t}t} \right)}{\pi \; t}\mspace{14mu} {and}\mspace{14mu} {\delta_{nT}(t)}} = {{\delta \left( {t - {nT}} \right)}.}}}$

a.) When T≦1/2Ω, then for tε

,

${f(t)} = {{T{\sum\limits_{n \in {\mathbb{Z}}}{{f({nT})}\frac{\sin \left( {\frac{\pi}{T}\left( {t - {nT}} \right)} \right)}{\pi \left( {t - {nT}} \right)}}}} = {{T\left( {\left\lbrack {\sum\limits_{n \in {\mathbb{Z}}}\delta_{nT}} \right\rbrack f} \right)}*\underset{T}{\sin \; c}{(t).}}}$

b.) When T≦1/2Ω and ƒ(nT)=0 for all nε

, then ƒ≡0.

We use the ON windowing systems to develop a sampling theory forultra-wide and adaptive bandwidth signals.

Again, some embodiments operate through a projection of the signal ontoblocks in time. For example, orthonormal (ON) windowing systems areprovided and developed to provide incorporate a sampling theory forultra-wide and adaptive bandwidth signals. Accordingly, the presentembodiments can represent changes of view in sampling, from that of astationary view of a signal used in classical sampling to, at least inpart, an adaptive windowed view or adaptive windowed stationary view.For example, this adaptive windowing, such as with at least some AFBcases, provides that the time and frequency space block or tile occupiedby the signal changes in time. The windows establish partitions of timeand/or frequency so that the signal can be sampled efficiently. Inultra-wide band (UWB) cases, for example, advantage is taken of thewindowing to partition the signal quickly and efficiently, and in someinstances uniformly. In some embodiments, within the blocks, the signalcan be sampled in parallel.

As introduced above, some embodiments provide partition of unity thatprovide for segmenting Time and/or Frequency (

-

) space. The partitioning provided can have variable partitioninglength, variable roll-off, and/or variable smoothness. The windows makesmooth adaptive partitions of unity, or bounded adaptive partitions ofunity systems. For example, these can be generalizations of boundeduniform partitions of unity in that they allow for signal adaptivewindowing. The construction elements for these methods and system areB-splines and take advantage of the concept of “perfect splineconstruction” (e.g., see I. J. Schoenberg, Cardinal Spline Interpolation(CBMS-NSF Conference Series in Applied Mathematics, 12), SIAM,Philadelphia, Pa., 1973, incorporated by reference). The B-splines givecontrol over smoothness in time and corresponding decay in frequency.The present embodiments can be configured to provide varying degrees ofsmoothness with cutoffs adaptive to signal information, e.g., bandwidth.

For example, in some embodiments, a straightforward system is created by{χ_([(k)T, (k+1)T])(t)}, for kε

. A second example can be developed by studying the de laVall'ee-Poussin kernel used in Fourier series (see T. W. Körner, FourierAnalysis, Cambridge University Press, Cambridge, 1988). Consider asignal block of length T+2r at the origin. Let 0<r<<T/2. Let

Tri_(L)(t)=max{[((T/(4r))+r)−|t|/(2r)], 0}.

Tri_(S)(t)=max{[((T/(4r))+r−1)−|t|/(2r)], 0} and

Trap_([−T/2−r, T/2+r])(t)=Tri_(L)(t)−Tri_(S)(t).  (1)

The Trap function can have perfect overlay in the time domain and 1/ω²decay in frequency space. When one time block is ramping down, anadjacent time block is ramping up at the same rate, and typically at theexact same rate. The system using overlapping Trap functions has theadvantage of 1/ω² decay in frequency. Let β_(L)=√{square root over(T/(4r)+r)}, α_(L)=T/(4r)+r/2, β_(S)=√{square root over (T/(4r)+r−1)},and α_(S)=T/(4r)−r/2. The Fourier transform of Trap can be:

$\begin{matrix}{{{Trap}{\,^{\hat{\;}}(\omega)}} = {\left\lbrack {\left( \beta_{L} \right)\frac{\sin \left( {2\pi \; \alpha_{L}\omega} \right)}{\pi \; \omega}} \right\rbrack^{2} - {\left\lbrack {\left( \beta_{S} \right)\frac{\sin \left( {2\pi \; \alpha_{S}\omega} \right)}{\pi \; \omega}} \right\rbrack^{2}.}}} & (2)\end{matrix}$

Further, some embodiments provide a bounded adaptive partition of unity.A bounded adaptive partition of unity is a set of functions {

_(k)(t)} such that:

-   -   (i.) supp(        _(k)(t))        [kT−r, (k+1)T+r] for all k.    -   (ii.)        _(k)(t))≡1 for tε[kT+r, (k+1)T−r] for all k,    -   (iii.) Σ        _(k)(t)|1    -   (iv.) {        [n]} is absolutely summable. i.e. {        [n]}εl¹.        Conditions (i.), (ii.) and (iii.) make {        _(k)(t)} a bounded partition of unity. Condition (iv.) provides        for the computation of Fourier coefficients.

For example, let ƒε

_(Ω) and {

_(k)(t)} be a bounded adaptive partition of unity with generating window

_(I). Let [ƒ]° be the T+2r periodization of ƒ. Then

$\begin{matrix}{{\frac{1}{T + {2r}}{\int_{{{- T}/2} - r}^{{T/2} + r}{\left\lbrack {f \cdot _{I}} \right\rbrack^{\circ}(t){\exp \left( {{- 2}\pi \; \; n\; {t/\left\lbrack {T + {2r}} \right\rbrack}} \right)}\ {t}}}} = {*{{\lbrack n\rbrack}.}}} & (4)\end{matrix}$

The above is supported as follows:

$\frac{1}{}\int_{}$

denote

$\frac{1}{T + {2r}}\int_{{{- T}/2} - r}^{{T/2} + r}$

and

denote [T+2r].

Then

$\begin{matrix}\begin{matrix}{{\hat{G}\lbrack n\rbrack} = {\frac{1}{}{\int_{}{\left\lbrack {f \cdot _{I}} \right\rbrack^{\circ}(t){\exp \left( {{- 2}\pi \; \; n\; {t/}} \right)}\ {t}}}}} \\{= {\frac{1}{}{\int_{}{{\left\lbrack {\sum\limits_{k}{{\lbrack k\rbrack}{\exp \left( {2\pi \; \; k\; {t/}} \right)}}} \right\rbrack \lbrack f\rbrack}\ {{^\circ}(t)}{\exp \left( {{- 2}\pi \; \; n\; {t/}} \right)}{t}}}}} \\{= {\sum\limits_{k}{{\lbrack k\rbrack}\frac{1}{}{\int_{}{\lbrack f\rbrack {{^\circ}(t)}{\exp \left( {{- 2}\; \pi \; {\left( {n\  - k} \right)}{t/}} \right)}{t}}}}}} \\{= {\sum\limits_{k}{{\lbrack k\rbrack}{\left\lbrack {n - k} \right\rbrack}}}} \\{= {*{{\lbrack n\rbrack}.}}}\end{matrix} & (5)\end{matrix}$

Examples:

-   -   {        _(k)(t)}=        χ_([(k)T, (k+1)T])(t)    -   {        _(k)(t)}=        Trap_([(k)T−r, (k+1)T+r])(t)

The above first example has jump discontinuities at segment boundariesof the blocks and has 1/ω² decay in frequency. The above second exampleis continuous, but typically not differentiable, and has overlaps atsegment boundaries of the blocks. This system has 1/ω² decay infrequency. Some embodiments generate systems by translations anddilations of a given window

_(I) where supp(

_(I))=[−T/2−r, T/2+r]. The generating window function

_(I) is k-times differentiable, has supp(

_(I))=[−T/2−r, T/2+r], and has values:

$\begin{matrix}{_{I} = \left\{ \begin{matrix}0 & {{t} \geq {{T/2} + r}} \\1 & {{t} \leq {{T/2} - r}} \\{\rho \left( {\pm t} \right)} & {{{T/2} - r} < {t} < {{T/2} + {r.}}}\end{matrix} \right.} & (6)\end{matrix}$

The ρ(t) can be solved by solving the Hermite interpolation problem:

$\quad\left\{ \begin{matrix}\left( {a.} \right) & {{\rho \left( {{T/2} - r} \right)} = 1} \\\left( {b.} \right) & {{{\rho^{(n)}\left( {{T/2} - r} \right)} = 0},{n = 1},2,{\ldots \mspace{14mu} k}} \\\left( {c.} \right) & {{{\rho^{(n)}\left( {{T/2} + r} \right)} = 0},{n = 0},1,2,{\ldots \mspace{14mu} {k.}}}\end{matrix} \right.$

with the conditions that ρεC^(k) and

[ρ(t))]+(ρ(−t))=1 for tε[T/2−r, T/2+r]  (7)

As described above, B-splines can be used. In some embodiments theB-splines are used as the cardinal functions. For example, let 0<α<<βand consider χ_([−α, α]). It is desirable, in at least some embodiments,that the n-fold convolution of χ_([α, α]) fit in the interval [−β, β].Further, α is chosen so that 0<nα<β, and let:

${\Psi (t)} = {\underset{n - {times}}{\underset{}{\chi_{\lbrack{{- \alpha},\alpha}\rbrack}*\chi_{\lbrack{{- \alpha},\alpha}\rbrack}*\mspace{11mu} \ldots \;*{\chi_{\lbrack{{- \alpha},\alpha}\rbrack}(t)}}}.}$

The β-periodic continuation of this function, Ψ°(t) has the Fourierseries expansion

$\sum\limits_{k \neq 0}{{\frac{\alpha}{n\; \beta}\left\lbrack \frac{\sin \left( {\pi \; k\; {\alpha/n}\; \beta} \right)}{2\pi \; k\; {\alpha/n}\; \beta} \right\rbrack}^{n}{{\exp \left( {\pi \; \; k\; {t/\beta}} \right)}.}}$

The C^(k) solution for p is given by a theorem of Schoenberg (see I. J.Schoenberg, Cardinal Spline Interpolation (CBMS-NSF Conference Series inApplied Mathematics, 12), SIAM, Philadelphia, Pa., 1973, pp. 7-8).Schoenberg solved the Hermite interpolation problem with endpoints −1and 1. An interpolant that minimizes the Chebyshev norm is called theperfect spline. The perfect spline S(t) for the Hermite problem withendpoints −1 and 1 such that

S(1)=1, S ^((n))(1)=0, n=1, 2, . . . , k, S ^((n))(−1)=0, n=0, 1, 2, . .. , k

is given by the integral of the function

${{M(x)} = {\left( {- 1} \right)^{n}{\sum\limits_{i = 0}^{k}\frac{\Psi \left( {t - t_{j}} \right)}{\varphi^{\prime}\left( t_{j} \right)}}}},$

where Ψ is the k−1 convolution of characteristic functions, the knotpoints are

${t_{j} = {- {\cos \left( \frac{\pi \; j}{n} \right)}}},$

j==0, 1, . . . , n and φ(t)Π_(j=0) ^(k)(t−t_(j)). When k is even, themidpoint occurs at the k/2 knot point. If k is odd, the midpoint occursat the midpoint between k/2 and (k+1)/2 knot points. Accordingly, thefollowing is provided:

${{\rho (t)} = {{Sol}(t)}},{{{where}\mspace{14mu} {l(t)}} = {{{- \frac{1}{r}}t} + {\frac{T}{2r}.}}}$

For this ρ, and for

$_{I} = \left\{ \begin{matrix}0 & {{t} \geq {{T/2} + r}} \\1 & {{t} \leq {{T/2} - r}} \\{\rho \left( {\pm t} \right)} & {{{T/2} - r} < {t} < {{T/2} + {r.}}}\end{matrix} \right.$

_(I)(ω) is given by the antiderivative of a linear combination offunctions of the form

$\left\lbrack \frac{\sin \left( {\pi \; k\; \alpha \; {\omega/n}\; T} \right)}{2\; \pi \; k\; \alpha \; {\omega/n}\; T} \right\rbrack^{n},$

and therefore has a decay of 1/ω^(n+1) in frequency.

As described above, some embodiments provide adaptive orthonormal (ON)windowing. These embodiments provide sets of windows so that theorthogonality of bases in adjacent and possible overlapping blocks ispreserved. These embodiments are provide, in at least someimplementations, based on solving a Hermite interpolation problem andenables control over smoothness in time and/or corresponding decay infrequency. In some embodiments, the systems implemented to providevarying degrees of smoothness with cutoffs adaptive to signal bandwidth.

Some system of signal segmentation use sine, cosine and linearfunctions. These embodiments can be relatively easy to implement, cancut down on frequency error and/or can preserve orthogonality. As anexample, a signal block of length T+2r centered at the origin can beconsidered. Let 0<r<<T/2. In some instances, it is desirable to minimizer as small as possible. The Cap(t) can be defined as follows:

$\begin{matrix}{{{Cap}(t)} = \left\{ \begin{matrix}0 & {{{t} \geq {{T/2} + r}},} \\1 & {{{t} \leq {{T/2} - r}},} \\{\sin \left( {{\pi/\left( {4r} \right)}\left( {t + \left( {{T/2} + r} \right)} \right)} \right)} & {{{{{- T}/2} - r} < t < {{{- T}/2} + r}},} \\{\cos \left( {{\pi/\left( {4r} \right)}\left( {t - \left( {{T/2} - r} \right)} \right)} \right)} & {{{T/2} - r} < t < {{T/2} + {r.}}}\end{matrix} \right.} & (8)\end{matrix}$

Given Cap, a blocking or tiling system {Cap_(k)(t)} can be formed suchthat supp(Cap_(k)(t))

[kT−r, (k+1)T+r] for all k. The Cap window has several properties thatmake it a good window for signal processing purposes. For example, theCap window has a partition property in that it windows or bounds thesignal in [−T/2−r, T/2+r] and is identically 1 on [−T/2+r, T/2−r].Additionally, the Cap window has a continuous roll-off at the endpoints,and has the property that for all tε

[Cap_(k)(t))]²+[Cap_(k+1)(t))]²=1

This last condition preserves the orthogonality of basis elementsbetween adjacent blocks. Additionally, it has 1/ω² decay in frequencyspace, and as a first time block is ramping down, an adjacent secondblock is ramping up at substantially if not exactly the same rate. Thesystem using overlapping Cap functions can have the additional advantageof 1/ω² decay in frequency. For example, letting T=2 and r=1:

$\begin{matrix}{{{Cap}^{\hat{\;}}(\omega)} = {\left\lbrack \frac{{\sin \left( {2\pi \; \omega} \right)} + {4\omega \; \cos \; \left( {4\pi \; \omega} \right)}}{\pi \; {\omega \left( {{16\omega^{2}} - 1} \right)}} \right\rbrack.}} & (9)\end{matrix}$

Again, let [ƒ]° be the T+2r periodization of ƒ. Because of both [f]° andCap have absolutely converging Fourier series,

${{\lbrack\rbrack}{{^\circ}\lbrack n\rbrack}} = {{\sum\limits_{m}{{\left\lbrack {n - m} \right\rbrack}{\lbrack m\rbrack}}} = {*{{{Cap}^{\hat{\;}}\lbrack n\rbrack}.}}}$

In theory the time domain may be cut up or blocked into perfectlyaligned segments so that there is no loss of information. Further, thesystems are configured to be smooth, so as to provide control over decayin frequency, have variable cut-off functions for flexibility in design,and adaptive, so as to adjust accordingly to changes in frequency band.Still further, the systems can be configured, in at least someembodiments, so that the orthogonality of bases in adjacent and possibleoverlapping blocks is preserved.

In some embodiments, an orthonormal (ON) window system may be configuredaccording to a set of functions {

_(k)(t)} such that for all kε

-   -   (i.) supp(        _(k)(t))        [kT−r, (k+1)T+r].    -   (ii.)        _(k)(t))|1 for tε[kT+r, (k+1)T−r],    -   (iii.)        _(k)((kT+T/2)−t)=        _(k)(t−(kT+T/2)) for tε[0, T/2+r].    -   (iv.) [        _(k)(t))]²+[        _(k+1)(t))]²=1.    -   (v.) {        [n]} is absolutely subbable, i.e. {        [n]}εI¹.

Conditions (i.) and (ii.) are partition properties, in that they give asnapshot of the input function ƒ on [kT+r, (k+1)T−r], with smoothroll-off at the edges. Conditions (iii.) and (iv.) preserveorthogonality between adjacent blocks. Condition (v.) provides for thecomputation of Fourier coefficients.

Some embodiments provide systems by translations and dilations of agiven window

_(I), where supp(

_(I))=[−T/2−r, T/2+r].

FIG. 4 shows a graphical representation of a series of windows 412-414,in accordance with some embodiments. In this representation, the windowsare shows such that as a first window 412 is ramping down 416 anadjacent second window 413 is ramping up 418. Similarly, as the secondwindow 413 is ramping down 420, a third window 414 is ramping up 422.Subsequent windows can be adaptively configured over the signal.

Condition (v.) above gives, for fε

_(Ω) and {

_(k)(t)} an orthonormal window system with generating window

_(I); that

1 T + 2  r  ∫ - T / 2 - r T / 2 + r  [ f ·  I ]  °   ( t )  exp ( - 2  π      n   t / [ T + 2  r ] )    t =  * I  [ n ]. ( 11 )

Examples

-   -   {        _(k)(t)}=        χ_([(k)T, (k+1)T])(t)    -   {        _(k)(t)}=        Cap_([(k)T−r, (k+1)T+r])(t).        The first example has jump discontinuities at segment boundaries        of time blocks and has 1/ω decay in frequency. The second above        example is continuous but not differentiable, and has overlaps        at segment boundaries of the time blocks. Further, the second        system has 1/ω² decay in frequency. In some embodiments, general        window function        _(I) can be k-times differentiable, can have

supp(

_(I))=[−T/2−r, T/2+r], and

can have values

$\begin{matrix}{_{I} = \left\{ \begin{matrix}0 & {{t} \geq {{T/2} + r}} \\1 & {{t} \leq {{T/2} - r}} \\{\rho \left( {\pm t} \right)} & {{{T/2} - r} < {t} < {{T/2} + {r.}}}\end{matrix} \right.} & (12)\end{matrix}$

The ρ(t) can be solved for by solving the Hermite interpolation problem:

$\quad\left\{ \begin{matrix}\left( {a.} \right) & {{\rho \left( {{T/2} - r} \right)} = 1} \\\left( {b.} \right) & {{{\rho^{(n)}\left( {{T/2} - r} \right)} = 0},{n = 1},2,{\ldots \mspace{14mu} k}} \\\left( {c.} \right) & {{{\rho^{(n)}\left( {{T/2} + r} \right)} = 0},{n = 0},1,2,{\ldots \mspace{14mu} k},}\end{matrix} \right.$

with the conditions that ρεC^(k) and

[ρ(t))]²+[ρ(−t))]²=1 for [T/2−r]≦|t|≦[T/2+r].  (13)

The constraint (13) directs solutions, in some embodiments, to beexpressed in terms of sin(t) and cos(t). Therefore, some embodimentssolve this interpolation problem is through the method of undeterminedcoefficients. The below solution for ρ demonstrates that a C¹ window canbe provided. It can initially be assumed that

ρ(t)=A sin(B[T/2−t])+C, T/2≦t≦T/2+r.

Since ρ is C¹, ρ′(T/2+r)=0; and so AB cos(B[r])=0, given B=π/2r. Windowcondition (iv.) above gives that 2[ρ(t/2)]²=1, and so C=√{square rootover (2)}/2. Finally, ρ(T/2+r)=r)=0, and so A=√{square root over (2)}/2.

To extend ρ onto T/2−r≦t≦T/2, window condition (iv.) from above is againapplied, providing

${\rho (t)} = {{{\sqrt{\left\lbrack {1 - {\frac{1}{2}\left\lbrack {1 - {\sin \left( {\frac{\pi}{2r}\left( {\frac{T}{2} - t} \right)} \right)}} \right\rbrack}^{2}} \right\rbrack} \cdot \frac{T}{2}} - r} \leq t \leq {\frac{T}{2}.}}$

Further, using window conditions (ii.) and (iii.) from above providing:

$\begin{matrix}{{\rho (t)} = \left\{ \begin{matrix}{\frac{\sqrt{2}}{2}\left\lbrack {1 - {\sin \left( {\frac{\pi}{2r}\left( {\frac{T}{2} - t} \right)} \right)}} \right\rbrack} & {{\frac{- T}{2} - r} \leq t \leq {\frac{- T}{2}.}} \\\sqrt{\left\lbrack {1 - {\frac{1}{2}\left\lbrack {1 - {\sin \left( {\frac{\pi}{2r}\left( {t - \frac{T}{2}} \right)} \right)}} \right\rbrack}^{2}} \right\rbrack} & {\frac{- T}{2} \leq t \leq {\frac{- T}{2} + {r.}}} \\1 & {{\frac{- T}{2} + r} < t < {\frac{T}{2} - {r.}}} \\\sqrt{\left\lbrack {1 - {\frac{1}{2}\left\lbrack {1 - {\sin \left( {\frac{\pi}{2r}\left( {\frac{T}{2} - t} \right)} \right)}} \right\rbrack}^{2}} \right\rbrack} & {{\frac{T}{2} - r} \leq t \leq {\frac{T}{2}.}} \\{\frac{1}{\sqrt{2}}\left\lbrack {1 - {\sin \left( {\frac{\pi}{2r}\left( {t - \frac{T}{2}} \right)} \right)}} \right\rbrack} & {\frac{T}{2} \leq t \leq {\frac{T}{2} + {r.}}}\end{matrix} \right.} & (14)\end{matrix}$

Accordingly, with each degree of smoothness, an additional degree ofdecay in frequency may be obtained with some embodiments.

The orthogonality between time blocks is also considered with someembodiments, where orthonormal (ON) window systems {

_(k)(t)} are configured so that they preserve orthogonality of basiselement of overlapping blocks. Because of the partition properties ofthese systems, the orthogonality of adjacent overlapping blocks may bechecked. Some constructions involve the folding technique described byCoifman and Meyer (e.g., see R. Coifman and Y. Meyer, Remarques surl'analyse de Fourier a fenetre, CR Acad. Sci. Paris 312, 259-261, 1991).Further, in some embodiments, the systems are developed constructivelyby using spline theory. The construction can be considered based on howthe extension for a system of sines and cosines could be implemented. Insome embodiments, the odd reflections can be extended about the leftendpoint and the even reflections about the right.

Let {φ_(j)(t)} be an orthonormal basis for L²[−T/2, T/2]. Define

$\begin{matrix}{{(t)} = \left\{ \begin{matrix}0 & {{t} \geq {{T/2} + r}} \\\phi_{j\; {(t)}} & {{t} \leq {{T/2} - r}} \\{- {\phi_{j}\left( {{- T} - t} \right)}} & {{{{- T}/2} - r} < t < {{- T}/2}} \\{\phi_{j}\left( {T - t} \right)} & {{T/2} < t < {{T/2} + {r.}}}\end{matrix} \right.} & (15)\end{matrix}$

Taking into consideration the orthogonality of overlapping blocks,{Ψ_(k, j)}={

_(k{tilde over (φ)}{tilde over (j)})(t)} can be an orthonormal basis forL²(

). With {

[n]}εl¹.

_(I)εL²[−T/2−r, T/2+r], it can follow that:

∥Ψ_(k, j)∥₂=∥

_(I)∥₂∥{tilde over (φ)}{tilde over (φ_(j))}∥₂<∞

Accordingly, it can be shown that

Ψ_(k, j), Ψ_(k, m)

=δ_(K, m)·δ_(j, n). The partitioning properties of the windows allow forlimiting the checking to overlapping and adjacent windows. When k=m, thewindow can be considered as centered at the origin and the basis {tildeover (φ)}_(j). It can be shown that

_(I){tilde over (φ)}_(i),

_(I{tilde over (φ)}{tilde over (j)})

=δi, j. Computing provides:

〈  I  i ,  I  〉 = ∫ - T / 2 - r - T / 2  (  I  ( t ) ) 2  ϕ i ( - T - t )  ϕ j  ( - T - t )    t + ∫ - T / 2 - T / 2 + r  ( ( I  ( t ) ) 2 - 1 )  ϕ i  ( t )  ϕ j   ( t )    t + ∫ - T / 2T / 2  ϕ i  ( t )  ϕ j  ( t )   t + ∫ T / 2 - r T / 2  ( (  I ( t ) ) 2 - 1 )  ϕ i   ( t )  ϕ j  ( t )    t + ∫ T / 2 T / 2 +r  (  I  ( t ) ) 2  ϕ i  ( T - t )  ϕ j  ( T - t )    t . ( 16)

Since {φ_(j)} in an orthonormal basis, the third integral equals 1. Alinear change of variables t=−T/2−τ can be applied to the first integraland t=−T/2+τ to the second integral. Adding these two integrals togetherprovides:

∫₀ ^(r)[(

_(I)(T/2−τ))²+(

_(I)(τ−T/2))²−1]φ_(i)(−T/2+τ)φ_(j)(−T/2+τ)dτ

Conditions (iii.) and (iv.) of our windowing system provides that theexpression

[(

_(I)(T/2−τ))²+(

(τ−T/2))²−1]

equals zero, and therefore the above integral equals zero. Applying thelinear change of variables t=T/2−τ to the fourth integral and t=T/2+τ tothe fifth integral shows that these two integrals also sum to zero byessentially the same argument.

It is further verified that

_(k{tilde over (φ)}i),

_(l{tilde over (φ)}{tilde over (j)})

=δ_(k, l)·δ_(i, j). Again, the partitioning properties of the widowsallows for confirmation by check adjacent windows. The symmetry of theconstruction allows the checking of

⁻¹ and

₀, where the overlapping region tε[−r, r] are checked. Accordingly, thefollowing provides:

${\langle{_{- 1}{{\overset{\sim}{\phi}}_{i}._{0}}{\overset{\sim}{\phi}}_{j}}\rangle} = {0 + {\int_{- r}^{0}{\left( {_{- 1}(t)} \right){\phi_{i}(t)}\left( {_{0}(t)} \right)\left( {- {\phi_{j}\left( {- t} \right)}} \right){t}}} + {\int_{0}^{r}{\left( {_{- 1}(t)} \right){\phi_{i}\left( {- t} \right)}\left( {_{0}(t)} \right){\phi_{j}(t)}{{t}.}}}}$

Applying the linear change of variables t=−τ to the first integral andsubstituting the variable τ and adding provides:

∫₀ ^(τ)[−

⁻¹(−τ)

₀(−τ)+

⁻¹(τ)

₀(τ)]φ_(i)(−τ)φ_(j)(τ)dτ.

Condition (iii.) of the windowing system provides that the expression

[−

⁻¹(−τ)

₀(−τ)+

⁻¹(τ)

₀(τ)]

equals zero, and thus the integral equals zero. Combining these twocomputations shows that:

Ψ_(k, j), Ψ_(m, n)

=δ_(k, m)·δ_(j, n).

Further, it can be shown that {Ψ_(k, j} spans L) ²(

). Given a function ƒεL², the windowed element ƒ_(k)(t)=

_(k)(t)·ƒ(t) can be considered. First, the expansion in the window

_(I) symmetric to the origin is considered. Let ƒ_(I)=

_(I)(t)·ƒ(t). The {φ_(j)(t)} is an orthonormal basis for L²[−T/2, T/2].Given ƒ_(I), define

$\begin{matrix}{{{\overset{\_}{f}}_{I}(t)} = \left\{ \begin{matrix}0 & {{t} \geq {{T/2} + r}} \\{f_{I}(t)} & {{t} \leq {{T/2} - r}} \\{{f_{I}(t)} - {f_{I}\left( {{- T} - t} \right)}} & {{{{- T}/2} - r} < t < {{- T}/2}} \\{{f_{I}(t)} + {f_{I}\left( {T - t} \right)}} & {{T/2} < t < {{T/2} + {r.}}}\end{matrix} \right.} & (18)\end{matrix}$

Since ƒ _(I)εL²[−T/2, T/2], it may be expanded as:

$\sum\limits_{j = 1}^{\infty}{{\langle{{\overset{\_}{f}}_{I} \cdot \phi_{j}}\rangle}{{\phi_{j}(t)}.}}$

To extend this to L²[−T/2−r, T/2+r], the expansion is performedaccording to {{tilde over (φ)}{tilde over (φ_(j))}(t)}, getting where

$\begin{matrix}{{{\overset{\overset{\sim}{\_}}{f}}_{I} = {\sum\limits_{j = 1}^{\infty}{{\langle{{\overset{\_}{f}}_{I} \cdot \phi_{j}}\rangle}{{\overset{\sim}{\phi}}_{j}(t)}}}},{where}} & (19) \\{{\overset{\overset{\sim}{\_}}{f}}_{I} = \left\{ \begin{matrix}0 & {{t} \geq {{T/2} + r}} \\{f_{I}(t)} & {{t} \leq {{T/2} + r}} \\{{f_{I}(t)} - {f_{I}\left( {{- T} - t} \right)}} & {{{{- T}/2} - r} < t < {{{- T}/2} + r}} \\{{f_{I}(t)} + {f_{I}\left( {T - t} \right)}} & {{{T/2} - r} < t < {{T/2} + {r.}}}\end{matrix} \right.} & (20)\end{matrix}$

Accordingly, this construction preservers orthogonality between adjacentblocks.

Additionally, with ƒ be any function in L², the windowed elementf_(k)(t)=

_(k)(t)·ƒ(t) as considered. The construction above was repeated for thiswindow, showing that, for fixed k, {Ψ_(k, j)} spans L²([kT−r, (k+1)T+r])and preserves orthogonality between adjacent blocks on either side.Summing over all kε

gives that {Ψ_(k, j)} is an orthonormal basis for L²(

).

Taking advantage of the above described windowing systems, someembodiments provide projection sampling. For example, adaptive frequencyband (AFB) and ultra-wide-band (UWB) systems typically need eitherrapidly changing or very high sampling rates. These rates stress signalreconstruction in a variety of ways. For example, sub-Nyquist samplingcreates aliasing error, but error would also show up in truncation,jitter and amplitude, as computation is stressed. The W-K-S samplingdoes not have a way to accurately reconstruct the signal for sub-Nyquistsamples nor adjust the sampling rate for variable band-width signals.Accordingly, with previous sampling, truncation can lose the energy inthe lost samples, aliasing can introduce ambiguous information in thesignal, and extremely high sampling can increase the likelihood ofjitter error. Further, perturbations of sampling sets of ultra-widebandsignals can result in unstable sampling sets.

Some present embodiments, however, provide sampling for adaptivefrequency band and ultra-wide-band systems. Further, some of theseembodiments provide a projection method. For example, the following wasconsidered:

Let ƒ be a signal of finite energy in the Paley-Wiener class

_(Ω). For a block of time T, let

${f(t)} = {\sum\limits_{k \in {\mathbb{Z}}}{{f(t)}{{\chi_{\lbrack{{{(k)}T},{{({k + 1})}T}}\rbrack}(t)}.}}}$

With a given block ƒ_(k)(t)=ƒ(t)Ψ_([(k)T, (k+1)T])(t), the function canbe T-periodically continued, such that:

[ƒ_(k)]°(t)=[ƒ(t)χ_([(k)T, (k+1)T])(t)]°.

Expanding [ƒ_(k)]°(t) in a Fourier series, provides:

${{\left\lbrack f_{k} \right\rbrack {{^\circ}(t)}} = {\sum\limits_{n \in {\mathbb{Z}}}{{\lbrack n\rbrack}{\exp \left( {2\pi \; \; n\; {t/T}} \right)}}}},{where},{{\lbrack n\rbrack} = {\frac{1}{T}{\int_{{(k)}T}^{{({k + 1})}T}{{f(t)}{\exp \left( {{- 2}\; \pi \; \; n\; {t/T}} \right)}\ {{t}.}}}}}$

The original function ƒ is Ω band-limited. However, the truncated blockfunctions ƒ_(k) are not. Using the original Ω band-limit gives a lowerbound on the number of non-zero Fourier coefficients [{circumflex over(ƒ)}{circumflex over (ƒ_(k)]°)}[n] as follows:

${\frac{n}{T} \leq \Omega},{i.e.},{n \leq {T \cdot {\Omega.}}}$

Accordingly, N=┌T·Ω┐ is chosen, where ┌·┐ denotes a ceiling function.For this choice of N, the following is computed:

$\begin{matrix}{{f(t)} = {\sum\limits_{k \in {\mathbb{Z}}}{{f(t)}{\chi_{\lbrack{{{(k)}T},{{({k + 1})}T}}\rbrack}(t)}}}} \\{= {\sum\limits_{k \in {\mathbb{Z}}}{\left\lbrack {\left\lbrack f_{k} \right\rbrack {{^\circ}(t)}} \right\rbrack {\chi_{\lbrack{{{(k)}T},{{({k + 1})}T}}\rbrack}(t)}}}} \\{\approx {\sum\limits_{k \in {\mathbb{Z}}}{\left\lbrack {\sum\limits_{n = {- N}}^{n = N}{{\lbrack n\rbrack}{\exp \left( {2\pi \; \; n\; {t/T}} \right)}}} \right\rbrack {{\chi_{\lbrack{{{(k)}T},{{({k + 1})}T}}\rbrack}(t)}.}}}}\end{matrix}$

It is noted that for this choice of the standard (sines, cosines) basis,it can be, for a fixed value of N, adjusted to a large bandwidth Ω bychoosing small time blocks T. Also, after a given set of time blocks, anincrease or decrease in bandwidth Ω can be accommodated by againadjusting the time blocks, e.g., given an increase in Ω, decrease T, andvice versa. These adjustments may result in the quality of the signal,as expressed in the accuracy the representation of ƒ, depends on N, Ω,and T.

Accordingly, the basic projection formula in accordance with someembodiments is given as follows. Let ƒε

_(Ω) and let T be a fixed block of time. Then, for N=┌T·Ω┐,

$\begin{matrix}\begin{matrix}{{f(t)} \approx {f_{proj}(t)}} \\{= {\sum\limits_{k \in {\mathbb{Z}}}{\left\lbrack {\sum\limits_{n = {- N}}^{N}{{\lbrack n\rbrack}{\exp \left( {2\pi \; \; n\; {t/T}} \right)}}} \right\rbrack {{\chi_{\lbrack{{kT},{{({k + 1})}T}}\rbrack}(t)}.}}}}\end{matrix} & (21)\end{matrix}$

This can be shows as follows: again, let ƒε

_(Ω) fix a block T, and providing that:

${f(t)} = {\sum\limits_{k \in {\mathbb{Z}}}{{f(t)}{{\chi_{\lbrack{{k\; T},{{({k + 1})}T}}\rbrack}(t)}.}}}$

On each time block, periodically extend the function to a T-periodicfunction, getting

[ƒ_(k)]°(t)=[ƒ(t)χ_(kT, (k+1)T)(t)]°(t).

Since [ƒ_(k)]°(t) is a periodic function, it can be expanded intoFourier series

${{f_{k}{{^\circ}(t)}} = {\sum\limits_{n \in {\mathbb{Z}}}{{\lbrack n\rbrack}{\exp \left( {2\pi \; \; n\; {t/T}} \right)}}}},$

where the coefficients [{circumflex over (ƒ)}{circumflex over(ƒ_(k)]°)}[n] are defined by

${\lbrack n\rbrack} = {\frac{1}{T}{\int_{kT}^{{({k + 1})}T}{{f(t)}{\exp \left( {{- 2}\pi \; \; n\; {t/T}} \right)}{{t}.}}}}$

Due to the fact the ƒ is Ω band-limited, it can be estimated that thevalue of n for which [{circumflex over (ƒ)}{circumflex over(ƒ_(k)]°)}[n] is non-zero. At minimum, [{circumflex over (ƒ)}{circumflexover (ƒ_(k)]°)}[n] is non-zero if

$\begin{matrix}{{\frac{n}{T} \leq \Omega},{{or}\mspace{14mu} {equivalently}},{n \leq {T \cdot {\Omega.{Let}}}}} & (22) \\{N = {\left\lceil {T \cdot \Omega} \right\rceil.}} & (23)\end{matrix}$

Summing provides

$\begin{matrix}{{f(t)} \approx {f_{proj}(t)}} \\{= {\sum\limits_{k \in {\mathbb{Z}}}{\left\lbrack {\sum\limits_{n = {- N}}^{N}{{\lbrack n\rbrack}{\exp \left( {2\pi \; \; n\; {t/T}} \right)}}} \right\rbrack {{\chi_{\lbrack{{kT},{{({k + 1})}T}}\rbrack}(t)}.}}}}\end{matrix}$

It is now evident how this method approximates the signal. Unlike theShannon method which examined the function at specific points, then usedthose individual points to recreate the curve, the projection method inaccordance with at least some embodiments breaks the signal into timeblocks or windows and then approximates their respective periodicexpansions with a Fourier series. This process allows the system toindividually evaluate each piece and base its calculation on the neededbandwidth. The individual Fourier series are then summed, recreating aclose approximation of the original signal. It is noted that instead offixing T, the method allows for the fixing of any of the three variables(N, T and/or Ω) while allowing the other two to fluctuate. From thedesign point of view, in some implementations the easiest and mostpractical parameter to fix may be N. For situations in which thebandwidth does not need flexibility, it may be beneficial to fix Ωand/or T based on the equation N=[T·Ω]. However, if greater bandwidth Ωis need, shorter time blocks T may be selected.

The projection method can adapt to changes in the signal. For example,consider that the signal ƒ(t) has a band-limit Ω(t) that changes withtime. This change effects the time blocking τ(t) and the number of basiselements N(t). This may make the analysis more complicated, butdemonstrates at least some of the advantage of the projection methodprovided by some embodiments over conventional methods.

For example, during a given τ(t), let Ω(t)=max{Ω(t): tετ(t)}. For asignal ƒ that is Ω(t) band-limited, the value of n can be estimated forwhich [{circumflex over (ƒ)}{circumflex over (ƒ_(k)]°)}[n] is non-zero.At minimum, [{circumflex over (ƒ)}{circumflex over (ƒ_(k)]°)}[n] isnon-zero if:

$\begin{matrix}{{\frac{n}{T(t)} \leq {\overset{\_}{\Omega}(t)}},{{or}\mspace{14mu} {equivalently}},{n \leq {{\tau (t)} \cdot {{\overset{\_}{\Omega}(t)}.{Let}}}}} \\{{N(t)} = {\left\lceil {{\tau (t)} \cdot {\overset{\_}{\Omega}(t)}} \right\rceil.}}\end{matrix}$

For this choice of N(t), the basic adaptive projection formula isobtained.

The projection method also adapts to general orthonormal systems, muchas Kramer-Weiss extends sampling to general orthonormal bases. Forexample, let ƒ, {circumflex over (ƒ)}εL²(

) and ƒ have a variable but bounded bandlimit Ω(t). Let τ(t) be anadaptive block of time. Given τ(t), let Ω(t)=max{Ω(t): tετ(t)}. ThenN(t)=[τ(t)· Ω(t)], ƒ(t)≈ƒ_(φ)(t), where

$\begin{matrix}{{f_{\rho}(t)} = {\sum\limits_{k \in {\mathbb{Z}}}{\left\lbrack {\sum\limits_{n = {- {N{(t)}}}}^{N{(t)}}{{\lbrack n\rbrack}c^{({2\; \pi \; {{int}/\tau}}}}} \right\rbrack {{x_{\lbrack{{kT},{{({k + 1})}T}}\rbrack}(t)}.}}}} & (24)\end{matrix}$

Further, given a function ƒ such that ƒε

_(Ω), let T be a fixed time block. Define ƒ(t) and ƒ_(k)(t) as in thebeginning of the computation above. Now, let {φ_(n)} be a generalorthonormal system for L²[0, T], and let {φ_(n, k)(t)}={φ_(n)(t−kT)}.Since ƒε

_(Ω), there exists N=N(T, Ω) such that {circumflex over (ƒ)}_(k)[n]=

ƒ·φ_(n, k)

=0 for all n>N. In fact, let

$N = {{\max\limits_{n}{\langle{f,\phi_{n,k}}\rangle}} \neq 0.}$

Expanding in a Fourier series relative {φ_(n, k} gives)

$\begin{matrix}{{{f_{k}(t)} = {\sum\limits_{n \in {\mathbb{Z}}}{{\hat{f_{k}}\lbrack n\rbrack}{\phi_{n,k}(t)}}}},{{{where}\mspace{14mu} {\hat{f_{k}}\lbrack n\rbrack}} = {{\langle{f_{k},\phi_{n,k}}\rangle}.}}} & (25)\end{matrix}$

Summing over all blocks gives the following.

Let {φ_(n)} be a general orthonormal system for L²[0, T] and let{φ_(n, k)(t)}={φ_(n)(t−kT)}. Let N=N(T, Ω) be such that ƒ_(k)[n]=0 forall n>N. Therefore, ƒ(t)≈ƒ_(φ)(t) where

$\begin{matrix}{{f_{P}(t)} = {\sum\limits_{k = {- \infty}}^{\infty}{\left\lbrack {\sum\limits_{n = {- N}}^{N}{{\langle{f_{k},\phi_{n,k}}\rangle}{\phi_{n,k}(t)}}} \right\rbrack {{x_{\lbrack{{kT},{{({k + 1})}T}}\rbrack}(t)}.}}}} & (26)\end{matrix}$

Accordingly, given characteristics of the class of input signals, thechoice of basis functions used in the projection method can be tailoredto optimal representation of the signal or a desired characteristic inthe signal.

Further, some embodiments provide for the projection formula fororthonormal windowing. For example, let {

_(k)(t)} be a orthonormal window system, and let {Ψ_(k, j)} be anorthonormal basis that preserves orthogonality between adjacent windows.Let ƒε

_(Ω) be such that

ƒ·

_(k), Ψ_(n)

for all n>N. Then, ƒ(t)≈ƒ_(φ)(t), where

f P  ( t ) = ∑ k ∈ ℤ  [ ∑ n = - N N  〈 f · k , Ψ n , k 〉  Ψ n , k ( t ) ] . ( 27 )

Given the flexibility of the windowing systems, some embodiments furtherprovide for an adaptive projection system for the orthonormal windowsystems. Let ƒ, {circumflex over (ƒ)}εL²(

) and ƒ have a variable but bounded band-limit Ω(t). Let τ(t) be anadaptive block of time. Let {

_(k)(t)} be a ON window system with window size τ(t)+2r on the kthblock, and let {Ψ_(k, j)} be an orthonormal basis that preservesorthogonality between adjacent windows.

Given τ(t), let Ω(t)=max{Ω(t): ƒετ(t)}. Let N(t)=N(τ(t), Ω(t)) be suchthat

ƒ·

_(k), Ψ_(n, k)

=0. Then, ƒ(t)≈ƒ_(φ)(t), where

f P  ( t ) = ∑ k ∈ ℤ  [ ∑ n = - N  ( t ) N  ( t )  〈 f · k , Ψ n ,k 〉  Ψ n , k  ( t ) ] ( 28 )

Examples

{ Ψ kj } = { k  ϕ j ~  ( t ) } .  where   { k  ( t ) } = ⋃ k ∈ ℤ χ [ ( k )  T  ( k + 1 )  T ]  ( t )   and { ϕ j ~ } = {  ( 2  π  n T  ( t - kT ) ) .  ( 2  π   n T  ( t - kT ) ) } .  { Ψ kj }= { k  ϕ j ~  ( t ) } .   where   { k  ( t ) } = ⋃ k ∈ ℤ  Cap [( k )  T - r · ( k + 1 )  T + r ]  ( t )${{and}\left\{ \overset{\sim}{\phi_{j}} \right\}} = {\left\{ {{\left( {\frac{2\pi \; n}{T}\left( {t - {kT}} \right)} \right).}\left( {\frac{2\pi \; n}{T}\left( {t - {kT}} \right)} \right)} \right\}.}$

The first example has jump discontinuities at segment boundaries and has1/ω decay in frequency. The second example is continuous but notdifferentiable, and has overlaps at segment boundaries. This system has1/ω² decay in frequency. The development of a C¹ system involves solvinga Hermite interpolation problem for not only the window but also thefolded basis elements. Using undetermined coefficients ρ can be solvedfor so that the window is C¹, getting

${\rho (t)} = \left\{ \begin{matrix}\sqrt{\left\lbrack {1 - {\frac{1}{2}\left\lbrack {1 - {\sin \left( {\frac{\pi}{2r}\left( {\frac{T}{2} - t} \right)} \right)}} \right\rbrack}^{2}} \right\rbrack} & {{\frac{T}{2} - r} \leq t \leq {\frac{T}{2}.}} \\{\frac{1}{\sqrt{2}}\left\lbrack {1 - {\sin \left( {\frac{\pi}{2}\left( {t - \frac{T}{2}} \right)} \right)}} \right\rbrack} & {{\frac{T}{2} \leq t \leq {\frac{T}{2} + r}},}\end{matrix} \right.$

Use the same technique can be used to solve for C¹ folded basis elements{{tilde over (φ)}{tilde over (φ_(j))}}. The constraints that make C¹folded basis elements are

$\begin{matrix}\left\{ \begin{matrix}\left( {a.} \right) & {{\phi_{j}\left( {{- T}/2} \right)} = 0} \\\left( {b.} \right) & {{\phi_{j}^{\prime}\left( {{- T}/2} \right)}\mspace{14mu} {exists}} \\\left( {c.} \right) & {{\phi_{j}^{\prime}\left( {T/2} \right)} = 0}\end{matrix} \right. & (29)\end{matrix}$

The constraint (29) can direct to solutions expressed in terms of sin(t)and cos(t). Solving the constraints (29) for φ_(j), provides:

$\begin{matrix}{{\phi_{j}(t)} = {\sqrt{\frac{2}{T}}{\sin \left( {{\pi \left( {k + {1/2}} \right)}\frac{\left( {t + {T/2}} \right)}{T}} \right)}}} & (30)\end{matrix}$

Example: A C¹ system

-   -   {Ψ_(k, j)}={        _(k{tilde over (φ)}{tilde over (j)})(t)}, where

l = { 0  t  ≥ T / 2 + r 1  t  ≤ T / 2 - r ρ  ( ± t ) T / 2 - r < t  < T / 2 + r ,   with   ρ  ( t ) = { [ 1 - 1 2  [ 1 - sin  ( π2  r  ( T 2 - t ) ) ] 2 ] T 2 - r ≤ t ≤ T 2 . 1 2  [ 1 - sin  ( π 2 r  ( t - T 2 ) ) ] T 2 ≤ t ≤ T 2 + r .   and   ϕ j  ( t ) = 2 T sin  ( π  ( k + 1 / 2 )  ( t + T / 2 ) T ) .

The computations become increasingly complex as the parameter kincreases. Accordingly, some embodiments provide for an “almostorthogonal” windowing systems using B-spline constructions. TheseB-spline constructions allow for a direct computation of the Fouriercoefficients.

The analysis of the error generated by the projection method involveslooking at the decay rates of the Fourier coefficients. Working with thestandard basis, for ƒεC(

_(2Φ)), the modulus of continuity can be defined as:

${{\mu (\delta)} = {\sup\limits_{{{x - y}} \leq \delta}{{{f(x)} - {f(y)}}}}},$

and have that

${{\hat{f}\lbrack n\rbrack}} \leq {\frac{1}{2}{{\mu \left( {1/n} \right)}.}}$

We say that ƒ satisfies a Hölder condition with exponent a when thereexists a constant K such that

[ƒ(x+δ)−ƒ(x)]≦Kδ ^(α).

When ƒ is k-times continuously differentiable and ƒ^(k) satisfies aHölder condition with exponent α, then there exists a constant K suchthat

${{\hat{f}\lbrack n\rbrack}} \leq {K{\frac{1}{n^{k + \alpha}}.}}$

The sharp cut-offs χ_([kT, (k+1)T]) have a decay of only 1/ω infrequency. The orthonormal ON windowing systems can be designed so thatthe windows have decay 1/(ω)^(k+2) in frequency. Thus, this can make theerror on each block summable.

Assuming

_(k) is C^(k), then

˜1/(ω^(k+2). The error ε_(k) _(φ) on a given block can further beanalyzed. Let M=∥(ƒ·

_(k))∥L²(

). Then

ɛ kp =  sup   ( f  ( t ) · ) - [ ∑ n = - N N  〈 f · k , Ψ n , k 〉 Ψ n , k  ( t ) ]  =  sup  [ ∑  n  > N  〈 f · k , Ψ n , k 〉 Ψ n , k  ( t ) ] ≤  [ ∑  n  > N  M n k + 2 ] .

The projection method according to at least some embodiments can beapplied with binary signals. For example, the Walsh functions {Υ_(n)}form an orthonormal basis for L²[0, 1]. The basis functions have therange {1, −1}, with values determined by a dyadic decomposition of theinterval. The Walsh functions are of modulus 1 everywhere. The functionsare given by the rows of the unnormalized Hadamard matrices, which aregenerated recursively by

${H(2)} = \begin{bmatrix}1 & 1 \\1 & {- 1}\end{bmatrix}$${H\left( 2^{({k + 1})} \right)} = {{{H(2)}{H\left( 2^{k} \right)}} = \begin{bmatrix}{H\left( 2^{k} \right)} & {H\left( 2^{k} \right)} \\{H\left( 2^{k} \right)} & {- {H\left( 2^{k} \right)}}\end{bmatrix}}$

It is noted that although the rows of the Hadamard matrices give theWalsh functions, the elements have to be reordered into sequency order.The components are typically arranged in ascending order of zerocrossings (see for example K. G. Beauchamp, Applications of Walsh andRelated Functions, Academic Press, London, 1984). The Walsh functionscan also be interpreted as the characters of the group G of sequencesover

₂. i.e., G=(

₂

. The Walsh basis is a well-developed system for the study of a widevariety of signals, including binary. The projection method according tosome present embodiments works with the Walsh system to create awavelet-like system to do signal analysis.

First assume that the time domain is covered by a uniform block tilingχ_([kT, (k+1)T])(t). The function can be translated and/or scaled onthis kth interval back to [0, 1] by a linear mapping. Denote theresultant mapping as ƒ_(k), which is an element of L²[0, 1]. Given thatƒε

_(Ω), there exists an N>0 (N=N(Ω)) such that

ƒ_(k), Υ_(n)

0 for all n>N. The decomposition of ƒ_(k) into Walsh basis elements is

$\sum\limits_{u = 0}^{N}{{\langle{f_{k},\mathrm{\Upsilon}_{n}}\rangle}{\mathrm{\Upsilon}_{n}.}}$

Translating and summing up gives a projection representation ƒ_(φ):

$\begin{matrix}{{f_{}(t)} = {\sum\limits_{k \in Z}^{\;}{\left\lbrack {\sum\limits_{n = 0}^{N}{{\langle{f_{k},\mathrm{\Upsilon}_{n}}\rangle}\mathrm{\Upsilon}_{n}}} \right\rbrack {{\chi_{\lbrack{{kT},{{({k + 1})}T}}\rbrack}}^{(t)}.}}}} & (31)\end{matrix}$

The windowing system results in limited or substantially no loss ofsignal data and orthogonality between signal blocks. Similarly, anorthonormal window system may be used with fixed T and/or adaptive twindow length. Again, the function ƒ·

_(k)(t) can be translated and/or scaled on this kth window back to [0,1] by a linear mapping. The resultant mapping can be denoted as ƒ_(k)_(τ) . The resultant function is an element of L²[0, 1]. Given that ƒε

_(Ω), there exists an M>0 (M=M(Ω)) such that

ƒ_(k) _(τ) , Υ_(n)

=0 for all n>M. The decomposition of ƒ_(k) _(τ) into Walsh basiselements is

$\sum\limits_{n = 0}^{M}{{\langle{f_{k},\mathrm{\Upsilon}_{n}}\rangle}{\mathrm{\Upsilon}_{n}.}}$

Again, translating and summing up gives the projection representationƒ_(φ) _(τ)

$\begin{matrix}{{f_{_{W}}(t)} = {\sum\limits_{k \in Z}^{\;}{\left\lbrack {\sum\limits_{n = 0}^{N}{{\langle{f_{k},\mathrm{\Upsilon}_{n}}\rangle}\mathrm{\Upsilon}_{n}}} \right\rbrack {{_{k}(t)}.}}}} & (32)\end{matrix}$

Almost Orthogonal Systems

Some embodiments simply provide processing by almost maintaining theorthogonality between windows. The partition of unity systems do notpreserve orthogonality between blocks when implementing the almostorthogonality. However, they are typically easier to compute andtypically easier to build in circuitry and/or ASICs. Therefore, thesesystems can be used to approximate the Cap system with B-splines. Usingthese approximations, windowing systems can be provided that nearlypreserve orthogonality. In many instances, each added degree ofsmoothness in time can add to the degree of decay in frequency.

The concept of almost maintaining orthogonality allows some embodimentsto create windowing systems that are more computable, and in someinstance can be more easily constructed or implemented (e.g., throughcircuitry, hardware and/or software), such as the Bounded AdaptivePartition of Unity systems {

_(k)(t)} with the orthogonality preservation of the ON Window System {

_(k)(t)}. For example, {

_(k)(t)}=

Cap_([(k)T−r, (k+1)T+r])(t) was considered, where

${{Cap}_{I}(t)} = \left\{ \begin{matrix}0 & {{t} \geq {{T/2} + {r.}}} \\1 & {{t} \leq {{T/2} - {r.}}} \\{\sin \left( {{\pi/\left( {4r} \right)}\left( {t + \left( {{T/2} + r} \right)} \right)} \right)} & {{{{- T}/2} - r} < t < {{{- T}/2} + {r.}}} \\{\cos \left( {{\pi/\left( {4r} \right)}\left( {t - \left( {{T/2} - r} \right)} \right)} \right)} & {{{T/2} - r} < t < {{T/2} + {r.}}}\end{matrix} \right.$

Let 0<r<<T, an almost orthonormal (ON) System for adaptive andultra-wide band sampling in accordance with some embodiments provides aset of functions {

_(k)(t)} for which there exists δ, 0≦δ≦½, such that

-   -   (i.) supp(        _(k)(t))        [kT−r, (k+1)T+r] for all k.    -   (ii.)        _(k)(t))≡1 for tε[kT+r, (k+1)T−r] for all k.    -   (iii.)        _(k)((kT+T/2)−t)=        _(k)(t−(kT+T/2)), tε[0, T/2+r].    -   (iv.) 1−δ≦[        _(k)(t))]²+[        _(k+1)(t))]²≦1+δ.    -   (v.){        [n]}εl¹.        Accordingly, some embodiments start with        Cap_([(k)T−r, (k+1)T+r])(t). These embodiments can place        equidistant knot points −T/2−r<t<−T/2+r and T/2−r<t<T/2+r, and        approximate sin, cos in those intervals with C^(k) B-splines.        For these systems, δ→0 as k increases. The partition of unity        systems typically do not preserve orthogonality between blocks        relative to the almost orthogonality embodiments. However, they        are typically easier to compute and they can be easier to build        in circuitry. Therefore, these systems can be used to        approximate the Cap system with B-splines. These embodiments can        provide windowing systems that nearly preserve orthogonality.        Each added degree of smoothness in time adds to the degree of        decay in frequency.

As such, the present embodiments provide windowing systems fortime-frequency analysis that can have variable partitioning length,variable roll-off and/or variable smoothness. This variability isdistinct from other standard sampling techniques, such as the standardShannon (W-K-S) sampling that effectively provides a signal stationarypoint-of-view. FIG. 5 shows an example sampling technique in accordancewith a standard Shannon (W-K-S) sampling. This sampling represents asignal stationary point-of-view sampling that is limited to a fixedsampling rate with rigid boundaries.

Referring back to FIG. 2, some embodiments are configured to provide fora projection method of processing a signal. A signal is initiallywindowed in step 212. Again, the windowing can be provided through anadaptive windowing.

FIGS. 6A-6B show simplified graphical representations of a method ofprojection sampling in accordance with some embodiments. Referring toFIG. 6A, a signal s(t) 610 is received, which can have variablebandwidth over the duration of the signal. For example, the signal s(t)610 can be similar to the signal 310 of FIG. 3A, and can include a highfrequency burst 612. As described above, the projection samplingincludes the partitioning of the signal into a plurality of windows(s(t). ′

_(k)(t)), such as ′

₁, ′

₂, ′

₃ . . . ′

_(k). In some embodiments, the partitioning can provide for adaptivewindows. where the windows can be adaptive based on N=[T·Ω], which canin at least some instances accommodate the variations in the signalbandwidth.

The windows are processed to perform a transform series expansions, forexample, through Fourier series (̂s) 616 providing transform coefficients618 (s·W_(k)

Ψ_(n, k)) in the frequency domain relative to the respective windows andin accordance with the number of basis elements (N) corresponding to therespective window (k). As described above, in some implementations thenumber of basis elements N is defined according to a ceiling based onT·Ω (i.e., N=⇄T·Ω┌). The windows can then be adaptive, for example,based on bandwidth Ω. As further described above, however, any one ofthe variables (N, T, Ω) may be varied in adapting the windows. Tosimplify implementation, for example through a system and/or circuitry,some embodiments may fix N.

It is noted that the processing to provide the expansion may beperformed in parallel. In some embodiments, as described above, thetransform coefficients (e.g., the Fourier coefficients) for a window ′

_(k) can be generated in parallel. Further, the expansion can maintainorthogonality of adjacent windows, including the orthogonality foroverlapping regions of adjacent windows at least in part through theproduct of the windowed signal s·′

_(k) with Ψ_(n, k).

Referring to FIG. 6B, a projection of the coefficients 630 is shownrelative to each window. An analysis is performed, in part, through asummation

$\left( {\sum\limits_{n = 1}^{N}{C_{({n,k})}\Psi_{n,k}}} \right)$

of the individual Fourier series 622 in accordance with

${{s(t)} \approx {\sum\limits_{k}^{\;}\left\lbrack {\sum\limits_{n = 1}^{N}{{\langle{{s \cdot W_{k}},\Psi_{n,k}}\rangle}\Psi_{n,k}}} \right\rbrack}},$

(see equations 26 and 27 above). In some embodiments, this can providean recreation of a close approximation of the original signal.

Referring back to FIG. 2, again some embodiments provide the projectedsampling based on the adaptive signal windowing 212, a transmission oranalysis 214 of the signal and a reconstruction or synthesis of thesignal 216. FIG. 7 shows a simplified graphical representation of atleast portions of a series of windows 612-614 (′

_(k), ′

_(k+1), ′

_(k+2), etc.), in accordance with some embodiments, provided in responseto the windowing in step 212. Some embodiments implement a number ofseparate processings to perform the windowing. Further, in someinstances, the separate processing can be performed by separatecircuitry and/or chips. In the example in FIG. 7, there are threeseparate processings, where in some embodiments the processing can beperformed by three separate processes and/or separate circuitry orchips. In this example, processing is being performed for the firstwindow ′

_(k) 612, while preparing to initiate processing for the second window ′

_(k+1) 613, and while processing for the third window ′

_(k+2) is idle or in a rest state. This rest state can be advantageousin some implementations as it can allow circuitry to cool down to limitor avoid overheating of the circuitry, which can cause non-linearoperation of the circuitry.

As window 1 begins to ramp down, the processing relative to the signalcorresponding to window 2 begins ramping up, while processingcorresponding to window 3 can be idle. Similarly, as window 2 beginsramping down, the processing of the signal corresponding to window 3ramps up, while processing corresponding to window 1 (i.e., a subsequentwindow following window 3 in time) is idle. This provides a cycling ofprocessing, and in this example, provides a three phase cycling (modulo3). Other cycling can be provided, such as a five phase cycle when itwould be beneficial, such as providing greater idle or cool down time.

The multiple modulo implementation can further be advantageous, forexample, in that when ′

_(k) is ramping down, and ′

_(k+1) is ramping up; processing would have to effectively wrap ′

_(k+1) to simultaneously provide processing for ′

_(k+2). Alternatively, with the plurality of processing phases, while ′

_(k+1) is in its flat portion ′

_(k) can be off, and ′

_(k+2) is getting ready to activate. Then processing can subsequentlyreconnect with ′

_(k+2) while ′

_(k+2) is in its flat spot, hence the modulo 3 and the three basewindows.

FIG. 8 shows a simplified flow diagram of a process 810 of providing thetransmission and/or analysis of the windowed signal (s(t)·′

_(k)(t)), in accordance with some embodiments, and in some instances canbe used to implement step 214 of the process 210 of FIG. 2, and in someembodiments, corresponds to the processing of a windowed signalrepresented in FIGS. 6A-6B. In step 812, the windowed signal is receivedand read in. In step 814, respective periodic series expansions areperformed on the window (k). This can provide a computable, atomictime-frequency decomposition of the signal, which can be sensitive toboth position in time and frequency simultaneously. For example, ageneralized Fourier series expansion can be applied in obtaining thecoefficients.

It is noted that in some embodiments, some or all of the expansion(and/or sampling, e.g., analog-to-digital sampling) can be performed inparallel, where multiple coefficients of a window, and typically all ofthe coefficients of a window can be processed in parallel. As describedabove, the parallel processing can be advantageous with ultra-wide band(UWB) signals to perform the expansion by constructed in parallel. Insome instances, the parallel processing may be achieved, for example, byfixing N across multiple windows over the signal, which can fix thenumber of base elements (and “Fourier coefficients”) that are computed.For example, an UWB signal can be mapped quickly using a fixed windowsize, where with each window the processing does not have to seriallywait for samples. Instead, the sampling coefficients (e.g., coefficients618 of FIG. 6A) for an entire window of the signal can be constructedsimultaneously in parallel in the frequency space. The parallelprocessing is typically quicker. Adaptive windowing can still bepreformed while utilizing parallel processing, but typically at theexpense of increased processing and/or processing time.

Still referring to FIG. 8, in step 816 the series coefficients (e.g.,Fourier coefficients) can be transmitted, stored and/or analyzed. Forexample, the sampling and the Fourier coefficients can be utilized as afirst part of a spectrum analyzer, where spectral analysis can beperformed on the Fourier coefficients. Further, in at least someembodiments the transmit, store and/or analysis of step 816 can be atleast partially implemented consistent with FIG. 6B based in theprojection of the coefficients 630 (C_((1, k)), C_((2, k)), . . . ,C_((N, k))).

FIG. 9 depicts a simplified flow diagram of a process 910 of providing areconstruction and/or synthesis of the coefficients in accordance withsome embodiments. In some embodiments, the process 910 can implementsome or all of step 216 of FIG. 2. In step 912, coefficients for atleast a pair of adjacent windows are received. In step 914, thesynthesis of overlapping windows is performed such that

${\sum\limits_{n = 1}^{N}{C_{({n,k})}{\Psi_{n,k}(t)}}} + {\sum\limits_{n = 1}^{N}{C_{({n,{k + 1}})}{{\Psi_{n,{k + 1}}(t)}.}}}$

Because of the previously performed analysis in the frequency domain theerrors on each window are summable, and the reconstruction through thesummation during the synthesis provides substantially a perfectreconstruction in the time domain. Accordingly, in some embodiments, thecoefficients are considered in pairs of adjacent windows, for example:

{C _((1, k)) , C _((2, k)) , . . . , C _((N, k))}, and

{C _(1, k+1)) , C _((2, k+1)) , . . . , C _((N, k+1))}.

This summation is not limited to just k and k+1, but is applicable toany k (arbitrary index k). Again, the use of the pair of coefficients isbased on the overlap between windows. The Ψ and C_(k) have built intothem the overlap, and the coefficients of adjacent windows C_(n, k) andthe C_(n, k+1) cooperate with each other perfectly (e.g., sine andcosine, or substantially any orthonormal basis), consistent with thefolding technique. Accordingly, from a summation point of view, theramps of the windows are effectively eliminated. Further, in someimplementations, the summation can split coefficients relative tooverlapping windows, while still summing in at least pairs of windows.

Still referring to FIG. 9, in step 916 a reindexing is performed (e.g.,k goes to k+1 and k+1 goes to k, in the formula, relative to asubsequent window to continue the window pairing. The process 910 canthen return to step 912 to continue the reconstruction and/or synthesisof the signal in the time domain over the remainder of the signal (oruntil the process is otherwise terminated).

In some implementations, the transform and computation of thecoefficients can be considered as analysis under step 214 of FIG. 2,while the synthesis in step 216 of FIG. 2 can comprise the re-buildingof the signal in the time domain. It is noted that at least a portion ofthe reconstruction and/or synthesis of step 216 and the process 910occurs in the time domain. Accordingly, the reconstruction and/orsynthesis can introduce truncation error. The amount of truncationerror, however, can be at least partially controlled and/or reduced byproviding smooth ramping windows, where the smoother the windows thesmaller the truncation error that occurs. It is further noted, however,the there is no jitter error at least in part due to the windowingand/or parallel processing. Similarly, there is typically no aliasingerror as a result of the above described processes because theretypically is no aliasing implemented.

In traditional analog-to-digital (A/D) and/or digital-to-analog (D/A)sampling there are four main types of errors: truncation error thattypically cut off samples in time, aliasing error where sample typicallycannot be performed quickly enough, jitter error in which incorrectplacement of the sample occurs, and computational error that isgenerally inherent error resulting from the systems implementing thesampling. Again, as described above, many if not all of the embodimentscan substantially eliminate two of the main errors. Generally, there isno aliasing performed and as such there is no aliasing error. Further,the windowing provided in the present embodiments, parallel processingand/or the pairing of adjacent windows can eliminate or substantiallyreduce jitter error. Accordingly, the errors result from truncationerrors and computation errors. The truncation errors can occur on eachwindow. These truncation errors, however, can be reduced through controlof the smoothness of the window partitions or boundaries, where thesmoother the window the less truncation error that occurs. Again,computational error is inherent in the system and can be reduced throughprecision design, manufacturing, implementation and/or assembly of thesystems, code and/or software used in implementing the presentembodiments

The present embodiments provide windowing methods and systems fortime-frequency analysis. At least some of these embodiments provide forwindowing with variable partitioning length, variable roll-off andvariable smoothness. Further, in some instances, the adaptive windowscan be constructed with smooth bounded adaptive partitions (which may beof unity) using B-splines. These methods and systems are useful whenevera partition of unity is used, such as in compressed sensing.

Some embodiments further preserve orthogonality of orthonormal systemsbetween adjacent windows. These are used to develop windowing systemsfor time-frequency analysis, and can provide a “projection method” fortime-frequency analysis of a signal. Still further, some embodimentssimplify and/or reduce processing through a concept of almostorthogonality and the B-spline techniques to create almost orthogonalwindowing systems that can often be more readily computable and/orconstructible through circuits and/or chips than the orthogonalitypreserving systems, which can result in lower costs and/or fasterprocessing while still providing results that satisfy desired thresholdaccuracy.

The projection method can comprise a method for analog-to-digitalencoding that can be implemented similar to or as an alternative toShannon Sampling. Further, some embodiments of the projection method canprovide accurate processing of adaptive frequency band (AFB) and/orultra-wide band (UWB) signals that typically cannot accurately beprocessed with traditional W-K-S Sampling. The present embodimentsprovide quick and accurate computations of Fourier coefficients, whichin some implementations can be computed in hardware, where at least someembodiments can be configured to implement the effective adaptivewindowing systems. For example, the computation of the coefficientsallow for very short (e.g., for UWB) and/or variable (e.g., for AFB)windows, and the design of the orthonormal (ON) windowing systemspreserve orthogonality between blocks and provide decay for themodulation of the signals caused by truncation in time. Given an ONwindow system {

_(k)(t)} and {Ψ_(k, j)}, an orthonormal basis that preservesorthogonality between adjacent windows, for ƒε

_(Ω), letting N=N(T, Ω) such that

ƒ·

_(k), Ψ_(n)

=0 for n>N; then, ƒ(t)≈ƒ_(φ)(t), where

${f_{}(t)} = {\sum\limits_{k \in Z}^{\;}{\left\lbrack {\sum\limits_{n = {- N}}^{N}{{\langle{{f \cdot _{k}},\Psi_{n}}\rangle}{\Psi_{n}(t)}}} \right\rbrack.}}$

Further, with the flexibility of the windowing systems, an adaptiveprojection system for ON windowing can be achieved. Given ƒ. {circumflexover (ƒ)}εL²)

) and ƒ having a variable but bounded band-limit Ω(t), with τ(t) beingan adaptive block of time, letting {

_(k)(t)} be a ON window system with window size τ(t)+2r on the kth blockand {Ψ_(k, j)} be an orthonormal basis that preserves orthogonalitybetween adjacent windows; given τ(t), with Ω(t)=max{Ω(t): tετ(t)}, andletting N(t)=N(τ(t), Ω(t)) be such that

ƒ·

_(k), Ψ_(n)

=0; then, ƒ(t)≈ƒ_(φ)(t), where

${f_{}(t)} = {\sum\limits_{k \in Z}^{\;}{\left\lbrack {\sum\limits_{n = {- {N{(t)}}}}^{N{(t)}}{{\langle{{f \cdot _{k}},\Psi_{n}}\rangle}{\Psi_{n}(t)}}} \right\rbrack.}}$

It is noted that this adaptable time segmentation may, in someinstances, make the analysis more complicated, but demonstrates at leastsome of the benefits the present embodiments provide over conventionalmeans. Further, at least some aspects of the projection methods can beconsidered, in some instances, as an “adaptive Gabor-type” system foranalysis in time-frequency. These methods and/or systems can beconfigured to provide either very short and/or variable windowing, withwindows created using the theory of splines. The correspondingmodulation terms are from an ON basis which preserves orthogonalitybetween adjacent blocks and can be tailored to the class of inputsignals analyzed. It was considered that if one looks at the constructfor binary signals using Walsh functions, one is reminded of Haarwavelets. Some embodiments provide what can be considered Walshprojection systems. These systems can be configured as an “adaptivewavelet” system with substantially no fixed underlying window size. Apotential drawback or price paid for achieving this adaptability can bethe giving up of the structure of the Gabor or wavelet systems.

In some instances, the above described theory of windows, definedaccording to mathematical structures, in which to express sampling viathe projection method. Many non-uniform sampling schemes could beexpressed in terms of this language of frames. Accordingly, someembodiments provide a computable atomic decomposition of time-frequencyspace. These embodiments can be configured to provide a way ofnon-uniformly windowing or tiling time and frequency so that when a thesignal has, for example, a burst of high-frequency information, themethods and/or systems can window quickly and efficiently in time andbroadly in frequency, whereas when the signal has a relativelylow-frequency segment, windowing can be defined broadly in time andefficiently in frequency.

The present methods and systems can address efficiency and cost issuesof broadband wireless and wire line transmission and reception. The needto transmit a much larger volume of information at lesser costs isimperative in the ever-expanding communications industry. These methodsand systems can allow signals to be processed through variouscommunications devices significantly times faster than existing systems.Further, the present embodiments provide a significant change of view insignal processing, which movies the process from one of a stationaryviewpoint to a short-term or adaptive windowed stationarity. There arenumerous applications for the present embodiments that can:

-   -   result in the development of low-cost transceivers;    -   allow for more throughput or better communication quality in        existing communication links;    -   contribute to the growing demand for ultra-sensitive electronic        equipment (e.g., warfare equipment in the defense arena);    -   allow for more communications of all types from all sources        processed per energy spent on communication; and    -   numerous other advantageous results.

Techniques for windowing are useful in developing the time-frequencyanalysis of functions. Some embodiments provide windowing systems thathave variable partitioning length, variable roll-off and variablesmoothness. For example, some embodiments can be configured to constructsmooth bounded adaptive partitions, which in some instances can be ofunity, using B-splines. These systems give a flexible adaptive partitionof unity of variable smoothness. Further, some embodiments can beconfigured to preserve orthogonality of orthonormal systems betweenadjacent windows. For example, these embodiments may be used to providetiling systems for time-frequency analysis, and give a “projectionmethod” for time-frequency analysis of a signal. Still further, someembodiments implement a method of almost orthogonality and the B-splinetechniques to create almost orthogonal windowing methods and systemsthat can be more computable and/or constructible in some instances thansome of the orthogonality preserving methods and/or systems.

Again, the present embodiments provide a computable atomic decompositionof time-frequency space. Some embodiments provide efficient methods toanalyze signals allowing for changing and/or ultra-wide frequency bands.Further, some embodiments provide non-uniform windowing time andfrequency so that when a signal has a variation in bandwidth thewindowing can be altered, such as when a signal has a burst ofhigh-frequency information the method and/or system can window quicklyand efficiently in time and broadly in frequency, whereas when thesignal has a relatively low frequency segment the method and/or systemcan window broadly in time and efficiently in frequency. Still further,the systems are readily implemented in circuitry.

FIG. 10, there is illustrated a system 1000 that may be used inprocessing signals in accordance with at least some embodiments. Thesystem 1000 can include a received and/or transceiver 1002, one or morecommunication links, paths, buses or the like 1004, and one or moreprocessing systems, chips or units 1006. The transceiver 1002 can beconfigured to receive the signal to be processed. The processing systems1006 can be substantially any circuitry, circuits, chips, ASICs and/orcombinations thereof that can implement the processing, which caninclude but is not limited to one or more of perform the windowing, thetransform series expansion, the calculations, summations, sampling,transmitting, storing, analyzing, reconstructing, synthesizing,transmitting and the like. Similarly, the processing system 1006 mayinclude one or more processors, microprocessors, central processingunits, logic, local digital storage, firmware and/or other controlhardware and/or software. As described above, in some instances,multiple phase cycling (e.g., three phase cycling, five phase cycling,etc.) may be implemented. As such, the system may include multipleprocessing systems 1006 to implement the multiple cycles.

The methods, techniques, systems, devices, services, and the likedescribed herein may be utilized, implemented and/or run on manydifferent types of devices and/or systems. Referring to FIG. 11, thereis illustrated a system 1100 that may be used for any suchimplementations, in accordance with some embodiments. One or morecomponents of the system 1100 may be used for implementing any system,apparatus, module, unit or device mentioned above or below, or parts ofsuch systems, apparatuses, modules, unit or devices, such as for exampleany of the above or below mentioned circuitry, chips, ASICs, systems,processing systems 1006, processors, and the like. However, the use ofthe system 1100 or any portion thereof is certainly not required.

By way of example, the system 1100 may comprise a controller orprocessor module 1112, memory 1114, one or more communication links,paths, buses or the like 1120, and in some instances a user interface1116. A power source or supply (not shown) is included or coupled withthe system 1100. The controller 1112 can be implemented through one ormore processors, microprocessors, central processing unit, logic, localdigital storage, firmware and/or other control hardware and/or software,and may be used to execute or assist in executing the steps of themethods and techniques described herein, and control various transforms,analysis, transmissions, storage, reconstruction, synthesis, windowing,measuring, communications, programs, interfaces, etc. The user interface1116, when present, can allow a user to interact with the system 1100and receive information through the system. In some instances, the userinterface 1116 may includes a display 1122, LEDs, audio output, and/orone or more user inputs 1124, such as keyboard, mouse, track ball, touchpad, touch screen, buttons, track ball, etc., which can be part of orwired or wirelessly coupled with the system 1100.

Typically, the system 1100 further includes one or more communicationinterfaces, ports, transceivers 1118 and the like allowing the system1100 to at least receive signals, which can be communicated wired orwirelessly over substantially any communication medium (e.g., over adistributed network, a local network, the Internet, communication link1120, other networks or communication channels with other devices and/orother such communications). Further the transceiver 1118 can beconfigured for wired, wireless, optical, fiber optical cable or othersuch communication configurations or combinations of suchcommunications.

The system 1100 comprises an example of a control and/or processor-basedsystem with the controller 1112. Again, the controller 1112 can beimplemented through one or more processors, controllers, centralprocessing units, logic, software and the like. Further, in someimplementations the controller 1112 may provide multiprocessorfunctionality.

The memory 1114, which can be accessed by the controller 1112, typicallyincludes one or more processor readable and/or computer readable mediaaccessed by at least the controller 1112, and can include volatileand/or nonvolatile media, such as RAM, ROM, EEPROM, flash memory and/orother memory technology. Further, the memory 1114 is shown as internalto the system 1110; however, the memory 1114 can be internal, externalor a combination of internal and external memory. The external memorycan be substantially any relevant memory such as, but not limited to,one or more of flash memory secure digital (SD) card, universal serialbus (USB) stick or drive, other memory cards, hard drive and other suchmemory or combinations of such memory. The memory 1114 can store code,software, executables, scripts, data, signals, samples, coefficients,programming, programs, media stream, media files, identifiers, log orhistory data, user information and the like.

One or more of the embodiments, methods, processes, approaches, and/ortechniques described above or below may be implemented in one or moreprocessor and/or computer programs executable by a processor-basedsystem. By way of example, such a processor based system may comprisethe processor based system 1100, a computer, an encoder, ananalog-to-digital converter, a player device, etc. Such a computerprogram may be used for executing various steps and/or features of theabove or below described methods, processes and/or techniques. That is,the computer program may be adapted to cause or configure aprocessor-based system to execute and achieve the functions describedabove or below. For example, such computer programs may be used forimplementing any embodiment of the above or below described steps,processes or techniques. As another example, such computer programs maybe used for implementing any type of tool or similar utility that usesany one or more of the above or below described embodiments, methods,processes, approaches, and/or techniques. In some embodiments, programcode modules, loops, subroutines, etc., within the computer program maybe used for executing various steps and/or features of the above orbelow described methods, processes and/or techniques. In someembodiments, the computer program may be stored or embodied on acomputer readable storage or recording medium or media, such as any ofthe computer readable storage or recording medium or media describedherein.

Accordingly, some embodiments provide a processor or computer programproduct comprising a medium configured to embody a computer program forinput to a processor or computer and a computer program embodied in themedium configured to cause the processor or computer to perform orexecute steps comprising any one or more of the steps involved in anyone or more of the embodiments, methods, processes, approaches, and/ortechniques described herein. For example, some embodiments provide oneor more computer-readable storage mediums storing one or more computerprograms for use with a computer simulation, the one or more computerprograms configured to cause a computer and/or processor based system toexecute steps comprising: receiving a communication signal; adaptivelypartitioning the signal in a time domain into a plurality of windows ofthe signal; transforming each of the windows of the signal producingrespective expansions in a frequency domain and obtaining respectivesamples of the windows of signal in the frequency domain; and mappingthe samples in the frequency domain back into the time domain.

The present embodiments provide methods and systems configured toprovide time-frequency analysis, including windowing systems providingsignal time-frequency analysis. For example, some embodiments providemethods of processing signals. These methods can comprise: receiving asignal; adaptively partitioning the signal in a time domain into aplurality of windows of the signal; and transforming each portion of thesignal of each windows producing respective expansions in a frequencydomain and analyzing and/or obtaining respective samples of therespective expansions in the frequency domain. Some embodiments furthermap the samples in the frequency domain back into the time domain.

Many of the functional units described in this specification have beenlabeled as systems, modules, units, etc., in order to more particularlyemphasize their implementation independence. For example, a system ormodule may be implemented as a hardware circuit comprising custom VLSIcircuits or gate arrays, off-the-shelf semiconductors such as logicchips, transistors, or other discrete components. A system and/or modulemay also be implemented in programmable hardware devices such as fieldprogrammable gate arrays, programmable array logic, programmable logicdevices or the like.

Some or all of the systems and/or modules may also be implemented insoftware for execution by various types of processors. An identifiedsystem and/or module of executable code may, for instance, comprise oneor more physical or logical blocks of computer instructions that may,for instance, be organized as an object, procedure, or function.Nevertheless, the executables of an identified module need not bephysically located together, but may comprise disparate instructionsstored in different locations which, when joined logically together,comprise the module and achieve the stated purpose for the module.

Indeed, a system or module of executable code could be a singleinstruction, or many instructions, and may even be distributed overseveral different code segments, among different programs, and acrossseveral memory devices. Similarly, operational data may be identifiedand illustrated herein within systems or modules, and may be embodied inany suitable form and organized within any suitable type of datastructure. The operational data may be collected as a single data set,or may be distributed over different locations including over differentstorage devices, and may exist, at least partially, merely as electronicsignals on a system or network.

While the invention herein disclosed has been described by means ofspecific embodiments, examples and applications thereof, numerousmodifications and variations could be made thereto by those skilled inthe art without departing from the scope of the invention set forth inthe claims.

1. A method of processing a signal, comprising: receiving acommunication signal; adaptively partitioning the signal in a timedomain into a plurality of windows of the signal; transforming each ofthe windows of the signal producing respective expansions in a frequencydomain and obtaining respective samples of the windows of signal in thefrequency domain while preserving orthogonality of basis elements in thewindows, including regions of overlap; and mapping the samples in thefrequency domain back into the time domain.
 2. The method of claim 1,wherein the transforming the windows of the signal comprisesindividually applying Fourier series to each of the windows of thesignal.
 3. The method of claim 1, wherein the adaptively partitioningthe signal in the time domain into the plurality of windows comprisesapplying B-splines in constructing the windows of the signal.
 4. Themethod of claim 1, wherein the adaptively partitioning the signal in thetime domain into the plurality of windows comprises controlling asmoothness in time and corresponding decay in frequency of each windowof the signal.
 5. The method of claim 1, wherein the adaptivelypartitioning the signal in the time domain into the plurality of windowscomprises partitioning the signal in the time domain into the pluralityof windows such that a plurality of the windows that are adjacent haveoverlapping segment boundaries.
 6. The method of claim 1, wherein theadaptively partitioning the signal in the time domain into the pluralityof windows comprises defining the partitions such that orthogonality issubstantially preserved for orthonormal (ON) system between adjacentwindows.
 7. The method of claim 1, wherein the adaptively partitioningthe signal in the time domain into the plurality of windows comprises 8.A method of processing a signal, comprising: receiving a signal;partitioning the signal in a time domain into a plurality of windows ofthe signal; and transforming each of the windows of the signal producingrespective expansions in a frequency domain, where for each window ofthe signal of the respective expansions are obtained through parallelprocessing obtaining in parallel respective samples of the windows ofsignal in the frequency domain.
 9. The method of claim 8, wherein thetransforming each of the windows comprises transforming each of thewindows such that sampling coefficients for an entire window isconstructed simultaneously in parallel in the frequency domain.
 10. Themethod of claim 8, further comprising: mapping the samples in thefrequency domain back into the time domain.
 11. The method of claim 10,wherein the partitioning the signal comprises adaptively partitioningthe signal such that each of the plurality of windows are partitioned asa function of bandwidth of the window.
 12. The method of claim 10,wherein the partitioning the signal comprises partitioning the signalaccording to a fixed window size of each of the plurality of windows.13. A method of processing a signal, comprising: processing a signal;partitioning the signal in a time domain into a plurality of windows ofthe signal; and transforming each of the windows of the signal producingrespective expansions in a frequency domain and obtaining respectivesamples of the windows of signal in the frequency domain whilepreserving orthogonality between at least two of the plurality ofwindows.
 14. The method of claim 13, wherein the partitioning the signalcomprises partitioning the signal such that the at least two of theplurality of windows have overlapping regions; and wherein thetransforming each of the windows of the signal comprises transformingeach of the windows of the signal while preserving the orthogonalitybetween the at least two of the plurality of windows includingpreserving the orthogonality in the overlapping regions.
 15. The methodof claim 14, wherein the partitioning the signal in the time domain intothe plurality of windows comprises applying B-splines in constructingthe windows of the signal.
 16. The method of claim 14, wherein thepartitioning the signal in the time domain into the plurality of windowsof the signal comprises adaptively partitioning the signal in the timedomain into the plurality of windows of the signal such that the windowsvary as a function of bandwidth.
 17. A method of processing a signal,comprising: receiving a communication signal; adaptively partitioningthe signal in a time domain into a plurality of windows of the signal,wherein the adaptively partitioning comprises applying B-splines inconstructing the windows of the signal; and transforming each of thewindows of the signal producing respective expansions in a frequencydomain and analyzing the transformed windows of the signal in thefrequency domain.
 18. The method of claim 17, wherein the transformingeach of the windows of the signal producing respective expansions in thefrequency domain comprises transforming each of the windows of thesignal while preserving orthogonality between at least two of theplurality of windows.
 19. The method of claim 18, wherein thepartitioning the signal comprises partitioning the signal such that theat least two of the plurality of windows have overlapping regions; andwherein the transforming each of the windows of the signal comprisestransforming each of the windows of the signal while preserving theorthogonality between the at least two of the plurality of windowsincluding preserving the orthogonality in the overlapping regions